The topic of wired remotes is something that I spent some time on with a Panasonic engineer at NAB. This engineer said that yes, you can in fact use a LANC controller with the CX350. With that assurance from him, I placed an order for one, along with a Varizoom VZ Rock PZFII for Canon. Better cancel the order for that Varizoom.
I really need that ability as much of my work is done with cranes, etc. and I need the ability to control iris, etc. in those situations.
The VariZoom VZ-Rock and VZ-Stealth (the LANC versions, not the Panasonic versions) do work with the CX350 and can remotely control zoom, focus, and start/stop recording. The only thing that isn't possible with these remotes is iris control.
As far as I know, the only way to remotely control iris on the camera currently is using the app.